Move from a pattern in words to a table, flow diagram, and formula—and use the formula to solve new problems.
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\[\text{words}\longleftrightarrow\text{flow}\longleftrightarrow\text{table}\longleftrightarrow\text{formula}\]
Introduction
A general rule tells you the term at any position. This lesson builds the bridge from an observed pattern to algebraic notation.
What is the main advantage of a formula?
\[T_n=3n+2\]
Topic 1 · Four representations
Multiply the position (n) by 3, then add 2. The result is the term (T_n).
\[n\longrightarrow\times3\longrightarrow+2\longrightarrow T_n\]
Topic 1 · Four representations
Operations are applied from left to right. Reversing them would produce a different rule.
\[\begin{array}{c|cccc}n&1&2&3&4\\\hline T_n&5&8&11&14\end{array}\]
Topic 1 · Four representations
For (n=1), (T_1=3(1)+2=5). Repeat for each position to generate the sequence.
\[\text{Multiply position by 3, then add 2}\]
Topic 1 · Four representations
The same order and operations appear in the verbal rule, flow diagram, table and formula.
\[\boxed{n\mapsto T_n\text{ through an operation rule}}\]
Topic 1 · Checkpoint
You can translate between words, a flow diagram, a table and a formula.
\[4,\ 7,\ 10,\ 13,\ldots\qquad d=3\]
Topic 2 · Linear rules
For a constant-difference pattern, begin with (dn). Here (d=3), so begin with (3n).
\[3n: 3,6,9,12\qquad\text{target}:4,7,10,13\]
Topic 2 · Linear rules
The target terms are always 1 greater than (3n). Add 1: (oxed{T_n=3n+1}).
\[T_n=T_1+(n-1)d\]
Topic 2 · Linear rules
For 4, 7, 10, 13, ... use (T_n=4+(n-1)(3)=3n+1).
\[8,12,16,20,\ldots\Rightarrow T_n=4n+4\]
Topic 2 · Linear rules
The difference is 4, but (4n) starts at 4. Add 4 so that (T_1=8).
\[\boxed{T_n=dn+\text{adjustment}}\]
Topic 2 · Checkpoint
You can use the constant difference as a coefficient, find the adjustment, and verify the first terms.
\[20,16,12,8,\ldots\qquad d=-4\]
Topic 3 · Decreasing rules
Start with (-4n). At (n=1), this gives (-4), but the target is 20.
\[-4n+24\Rightarrow20,16,12,8,\ldots\]
Topic 3 · Decreasing rules
Add 24. The rule is (oxed{T_n=-4n+24}). Check at positions 1 and 4.
\[30,25,20,15,\ldots\Rightarrow T_n=-5n+35\]
Topic 3 · Decreasing rules
The difference is (-5). Begin with (-5n), then add 35 to produce the first term 30.
\[T_n=25-5(n-1)=-5n+30\]
Topic 3 · Decreasing rules
The first-term form and simplified form are equivalent. Both give 25 at (n=1) and decrease by 5.
\[\boxed{d<0\Rightarrow\text{negative coefficient of }n}\]
Topic 3 · Checkpoint
You can write and check general rules for decreasing constant-difference patterns.
\[n\longrightarrow\times5\longrightarrow-2\longrightarrow T_n\]
Topic 4 · Flow and reverse
The flow diagram gives (T_n=5n-2). For (n=4), the output is (18).
\[T_n=23\Rightarrow 5n-2=23\]
Topic 4 · Flow and reverse
Undo (-2) by adding 2, then undo ( imes5) by dividing by 5. This gives (n=5).
\[23\xrightarrow{+2}25\xrightarrow{\div5}5\]
Topic 4 · Flow and reverse
The last forward operation is undone first. This is why the reverse path begins by adding 2.
\[T_{10}=7(10)-3=67\]
Topic 4 · Flow and reverse
When the position is known, substitute it directly into the rule. Keep brackets around the substituted value.
\[\boxed{n\to T_n:\text{ forward}\qquad T_n\to n:\text{ reverse}}\]
Topic 4 · Checkpoint
You can use a rule to find a term and reverse the operations to find a position.
\[3,6,11,18,27,\ldots\qquad T_n=n^2+2\]
Topic 5 · Non-linear rules
Subtract 2 from each term: 1, 4, 9, 16, 25. These are square numbers.
\[2,4,8,16,32,\ldots\qquad T_n=2^n\]
Topic 5 · Non-linear rules
The exponent is the position. At (n=5), (T_5=2^5=32).
\[4,9,16,25,36\Rightarrow T_n=(n+1)^2\]
Topic 5 · Non-linear rules
The terms are (2^2,3^2,4^2,5^2,6^2). The base is one more than the position.
\[n^2\neq2n\]
Topic 5 · Non-linear rules
(n^2=n imes n), while (2n=2 imes n). For (n=4), these give 16 and 8.
\[\boxed{\text{Do not force every pattern into }an+b}\]
Topic 5 · Checkpoint
You can recognise square, shifted-square and power rules and distinguish (n^2) from (2n).
\[\text{Rows: }12,16,20,24,\ldots\Rightarrow T_n=4n+8\]
Topic 6 · Apply and verify
A theatre starts with 12 seats in Row 1 and adds 4 per row. The formula is (12+4(n-1)=4n+8).
\[T_{18}=4(18)+8=80\]
Topic 6 · Apply and verify
Row 18 has 80 seats. Include the unit or object name in the final answer.
\[T_1=5,\quad T_2=14,\quad T_3=23\]
Topic 6 · Apply and verify
For (T_n=9n-4), these outputs agree with the sequence 5, 14, 23, .... A single match is not enough evidence.
How many positions should you check?
\[\text{rule}\rightarrow\text{substitute}\rightarrow\text{compare}\rightarrow\text{justify}\]
Topic 6 · Apply and verify
A justification names the pattern type, shows how the formula was built, and verifies several terms numerically.
\[\boxed{\text{A general rule predicts and explains}}\]
Topic 6 · Checkpoint
You can model a context, find distant terms, solve for a position, and justify a rule with multiple checks.
\[\begin{aligned}T_n&=T_1+(n-1)d\\&=dn+\text{adjustment}\end{aligned}\]
Summary
Represent the same relationship in words, a flow diagram, a table and a formula. For arithmetic patterns, the difference becomes the coefficient of (n); then determine the adjustment.
You are ready for the comprehensive pattern lesson.